PROPERTY (w) OF UPPER TRIANGULAR OPERATOR MATRICES

Let M-C = [GRAPHICS] is an element of L (X, Y) be be an upper triangulate Banach space operator. The relationship between the spectra of M-C and M-0, and their various distinguished parts, has been studied by a large number of authors in the recent past. This paper brings forth the important role played by SVEP, the single-valued extension property, in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type M-0 satisfies property (w) double left right arrow M-C satisfies property (w) to hold. Moreover, we explore certain conditions on T is an element of L (H) and S is an element of L (K) so that the direct sum T circle plus S obeys property (w), where H and K are Hilbert spaces.